Preheat oven to 350°F. Line two baking sheets with parchment paper.
Cream softened butter, brown sugar, and granulated sugar until light and fluffy, 2–3 minutes.
Beat in eggs one at a time, then mix in vanilla until smooth.
Whisk flour, cocoa, baking soda, salt, and espresso powder in a separate bowl.
Add dry ingredients to wet and mix on low just until combined. Do not overmix.
Fold in mini chocolate chips. Let dough rest 5 minutes to thicken.
Scoop 1½-tablespoon mounds, spacing 2 inches apart on the sheets.
Bake 9–11 minutes, until edges are set and centers look slightly soft.
Cool on the sheet 5 minutes, then transfer cookies to a rack to cool completely.
Warm cream in a small saucepan until steaming. Remove from heat; add chocolate chips and butter.
Let stand 2 minutes, then whisk until glossy and smooth to make ganache.
Spoon or spread ganache over cooled cookies and top with rainbow chips. Let set before serving.